Topological Expansion in the Complex Cubic Log-Gas Model. One-Cut Case

TL;DR

This paper proves the topological expansion for the partition function of a complex cubic log-gas model in the one-cut phase, using Riemann-Hilbert methods and quadratic differentials, revealing phase transitions and a tricritical point.

Contribution

It establishes the topological expansion for the complex cubic log-gas partition function in the one-cut phase, employing Riemann-Hilbert analysis and potential theory.

Findings

Topological expansion proven for the one-cut phase.

Identification of phase regions and critical arcs.

Connection to Painlevé I tricritical point.

Abstract

We prove the topological expansion for the cubic log-gas partition function \[ Z_N(t)= \int_\Gamma\cdots\int_\Gamma\prod_{1\leq j<k\leq N}(z_j-z_k)^2 \prod_{k=1}^Ne^{-N\left(-\frac{z^3}{3}+tz\right)}\mathrm dz_1\cdots \mathrm dz_N, \] where $t$ is a complex parameter and $\Gamma$ is an unbounded contour on the complex plane extending from $e^{\pi \mathrm i}\infty$ to $e^{\pi \mathrm i/3}\infty$. The complex cubic log-gas model exhibits two phase regions on the complex $t$-plane, with one cut and two cuts, separated by analytic critical arcs of the two types of phase transition: split of a cut and birth of a cut. The common point of the critical arcs is a tricritical point of the Painlev\'e I type. In the present paper we prove the topological expansion for $\log Z_N(t)$ in the one-cut phase region. The proof is based on the Riemann--Hilbert approach to semiclassical asymptotic expansions for the associated orthogonal polynomials and the theory of $S$-curves and quadratic differentials.